How do you use the definition of a derivative to find the derivative of # f(x) = -6 – x^2# at x=7?

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Answer 1 · 2018-01-27T06:39:16

At #x=7#, #(df)/(dx)=-14#

For a function #f(x)#, its derivative #(df)/(dx)# is defined as

#(df)/(dx)=lim_(h->0)(f(x+h)-f(x))/h#

As #f(x)=-6-x^2#, #f(x+h)=-6-(x+h)^2#

and #(df)/(dx)=lim_(h->0)(f(x+h)-f(x))/h#

= #lim_(h->0)(-6-(x+h)^2-(-6-x^2))/h#

= #lim_(h->0)(-6-x^2-2hx-h^2+6+x^2)/h#

= #lim_(h->0)(-2hx-h^2)/h#

= #lim_(h->0)(-2x-h)#

= #-2x#

and at #x=7#, we have #(df)/(dx)=-2xx7=-14#